\(\int \frac {a+b \text {arctanh}(c x^{3/2})}{x^2} \, dx\) [217]

Optimal result

Integrand size = 16, antiderivative size = 142 \[ \int \frac {a+b \text {arctanh}\left (c x^{3/2}\right )}{x^2} \, dx=-\frac {1}{2} \sqrt {3} b c^{2/3} \arctan \left (\frac {1-2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )+\frac {1}{2} \sqrt {3} b c^{2/3} \arctan \left (\frac {1+2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )+b c^{2/3} \text {arctanh}\left (\sqrt [3]{c} \sqrt {x}\right )-\frac {a+b \text {arctanh}\left (c x^{3/2}\right )}{x}+\frac {1}{2} b c^{2/3} \text {arctanh}\left (\frac {\sqrt [3]{c} \sqrt {x}}{1+c^{2/3} x}\right ) \] Output:

-1/2*3^(1/2)*b*c^(2/3)*arctan(1/3*(1-2*c^(1/3)*x^(1/2))*3^(1/2))+1/2*3^(1/ 
2)*b*c^(2/3)*arctan(1/3*(1+2*c^(1/3)*x^(1/2))*3^(1/2))+b*c^(2/3)*arctanh(c 
^(1/3)*x^(1/2))-(a+b*arctanh(c*x^(3/2)))/x+1/2*b*c^(2/3)*arctanh(c^(1/3)*x 
^(1/2)/(1+c^(2/3)*x))
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.44 \[ \int \frac {a+b \text {arctanh}\left (c x^{3/2}\right )}{x^2} \, dx=-\frac {a}{x}+\frac {1}{2} \sqrt {3} b c^{2/3} \arctan \left (\frac {-1+2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )+\frac {1}{2} \sqrt {3} b c^{2/3} \arctan \left (\frac {1+2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )-\frac {b \text {arctanh}\left (c x^{3/2}\right )}{x}-\frac {1}{2} b c^{2/3} \log \left (1-\sqrt [3]{c} \sqrt {x}\right )+\frac {1}{2} b c^{2/3} \log \left (1+\sqrt [3]{c} \sqrt {x}\right )-\frac {1}{4} b c^{2/3} \log \left (1-\sqrt [3]{c} \sqrt {x}+c^{2/3} x\right )+\frac {1}{4} b c^{2/3} \log \left (1+\sqrt [3]{c} \sqrt {x}+c^{2/3} x\right ) \] Input:

Integrate[(a + b*ArcTanh[c*x^(3/2)])/x^2,x]
 

Output:

-(a/x) + (Sqrt[3]*b*c^(2/3)*ArcTan[(-1 + 2*c^(1/3)*Sqrt[x])/Sqrt[3]])/2 + 
(Sqrt[3]*b*c^(2/3)*ArcTan[(1 + 2*c^(1/3)*Sqrt[x])/Sqrt[3]])/2 - (b*ArcTanh 
[c*x^(3/2)])/x - (b*c^(2/3)*Log[1 - c^(1/3)*Sqrt[x]])/2 + (b*c^(2/3)*Log[1 
 + c^(1/3)*Sqrt[x]])/2 - (b*c^(2/3)*Log[1 - c^(1/3)*Sqrt[x] + c^(2/3)*x])/ 
4 + (b*c^(2/3)*Log[1 + c^(1/3)*Sqrt[x] + c^(2/3)*x])/4
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.27, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {6452, 851, 754, 27, 219, 1142, 25, 27, 1082, 217, 1103}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {a+b \text {arctanh}\left (c x^{3/2}\right )}{x^2} \, dx\)

\(\Big \downarrow \) 6452

\(\displaystyle \frac {3}{2} b c \int \frac {1}{\sqrt {x} \left (1-c^2 x^3\right )}dx-\frac {a+b \text {arctanh}\left (c x^{3/2}\right )}{x}\)

\(\Big \downarrow \) 851

\(\displaystyle 3 b c \int \frac {1}{1-c^2 x^3}d\sqrt {x}-\frac {a+b \text {arctanh}\left (c x^{3/2}\right )}{x}\)

\(\Big \downarrow \) 754

\(\displaystyle 3 b c \left (\frac {1}{3} \int \frac {1}{1-c^{2/3} x}d\sqrt {x}+\frac {1}{3} \int \frac {2-\sqrt [3]{c} \sqrt {x}}{2 \left (c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1\right )}d\sqrt {x}+\frac {1}{3} \int \frac {\sqrt [3]{c} \sqrt {x}+2}{2 \left (c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1\right )}d\sqrt {x}\right )-\frac {a+b \text {arctanh}\left (c x^{3/2}\right )}{x}\)

\(\Big \downarrow \) 27

\(\displaystyle 3 b c \left (\frac {1}{3} \int \frac {1}{1-c^{2/3} x}d\sqrt {x}+\frac {1}{6} \int \frac {2-\sqrt [3]{c} \sqrt {x}}{c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}+\frac {1}{6} \int \frac {\sqrt [3]{c} \sqrt {x}+2}{c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}\right )-\frac {a+b \text {arctanh}\left (c x^{3/2}\right )}{x}\)

\(\Big \downarrow \) 219

\(\displaystyle 3 b c \left (\frac {1}{6} \int \frac {2-\sqrt [3]{c} \sqrt {x}}{c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}+\frac {1}{6} \int \frac {\sqrt [3]{c} \sqrt {x}+2}{c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}+\frac {\text {arctanh}\left (\sqrt [3]{c} \sqrt {x}\right )}{3 \sqrt [3]{c}}\right )-\frac {a+b \text {arctanh}\left (c x^{3/2}\right )}{x}\)

\(\Big \downarrow \) 1142

\(\displaystyle 3 b c \left (\frac {1}{6} \left (\frac {3}{2} \int \frac {1}{c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}-\frac {\int -\frac {\sqrt [3]{c} \left (1-2 \sqrt [3]{c} \sqrt {x}\right )}{c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}}{2 \sqrt [3]{c}}\right )+\frac {1}{6} \left (\frac {3}{2} \int \frac {1}{c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}+\frac {\int \frac {\sqrt [3]{c} \left (2 \sqrt [3]{c} \sqrt {x}+1\right )}{c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}}{2 \sqrt [3]{c}}\right )+\frac {\text {arctanh}\left (\sqrt [3]{c} \sqrt {x}\right )}{3 \sqrt [3]{c}}\right )-\frac {a+b \text {arctanh}\left (c x^{3/2}\right )}{x}\)

\(\Big \downarrow \) 25

\(\displaystyle 3 b c \left (\frac {1}{6} \left (\frac {3}{2} \int \frac {1}{c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}+\frac {\int \frac {\sqrt [3]{c} \left (1-2 \sqrt [3]{c} \sqrt {x}\right )}{c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}}{2 \sqrt [3]{c}}\right )+\frac {1}{6} \left (\frac {3}{2} \int \frac {1}{c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}+\frac {\int \frac {\sqrt [3]{c} \left (2 \sqrt [3]{c} \sqrt {x}+1\right )}{c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}}{2 \sqrt [3]{c}}\right )+\frac {\text {arctanh}\left (\sqrt [3]{c} \sqrt {x}\right )}{3 \sqrt [3]{c}}\right )-\frac {a+b \text {arctanh}\left (c x^{3/2}\right )}{x}\)

\(\Big \downarrow \) 27

\(\displaystyle 3 b c \left (\frac {1}{6} \left (\frac {3}{2} \int \frac {1}{c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}+\frac {1}{2} \int \frac {1-2 \sqrt [3]{c} \sqrt {x}}{c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}\right )+\frac {1}{6} \left (\frac {3}{2} \int \frac {1}{c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}+\frac {1}{2} \int \frac {2 \sqrt [3]{c} \sqrt {x}+1}{c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}\right )+\frac {\text {arctanh}\left (\sqrt [3]{c} \sqrt {x}\right )}{3 \sqrt [3]{c}}\right )-\frac {a+b \text {arctanh}\left (c x^{3/2}\right )}{x}\)

\(\Big \downarrow \) 1082

\(\displaystyle 3 b c \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1-2 \sqrt [3]{c} \sqrt {x}}{c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}+\frac {3 \int \frac {1}{-x-3}d\left (1-2 \sqrt [3]{c} \sqrt {x}\right )}{\sqrt [3]{c}}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {2 \sqrt [3]{c} \sqrt {x}+1}{c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}-\frac {3 \int \frac {1}{-x-3}d\left (2 \sqrt [3]{c} \sqrt {x}+1\right )}{\sqrt [3]{c}}\right )+\frac {\text {arctanh}\left (\sqrt [3]{c} \sqrt {x}\right )}{3 \sqrt [3]{c}}\right )-\frac {a+b \text {arctanh}\left (c x^{3/2}\right )}{x}\)

\(\Big \downarrow \) 217

\(\displaystyle 3 b c \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1-2 \sqrt [3]{c} \sqrt {x}}{c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}-\frac {\sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )}{\sqrt [3]{c}}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {2 \sqrt [3]{c} \sqrt {x}+1}{c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}+\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{c} \sqrt {x}+1}{\sqrt {3}}\right )}{\sqrt [3]{c}}\right )+\frac {\text {arctanh}\left (\sqrt [3]{c} \sqrt {x}\right )}{3 \sqrt [3]{c}}\right )-\frac {a+b \text {arctanh}\left (c x^{3/2}\right )}{x}\)

\(\Big \downarrow \) 1103

\(\displaystyle 3 b c \left (\frac {1}{6} \left (-\frac {\sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )}{\sqrt [3]{c}}-\frac {\log \left (c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1\right )}{2 \sqrt [3]{c}}\right )+\frac {1}{6} \left (\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{c} \sqrt {x}+1}{\sqrt {3}}\right )}{\sqrt [3]{c}}+\frac {\log \left (c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1\right )}{2 \sqrt [3]{c}}\right )+\frac {\text {arctanh}\left (\sqrt [3]{c} \sqrt {x}\right )}{3 \sqrt [3]{c}}\right )-\frac {a+b \text {arctanh}\left (c x^{3/2}\right )}{x}\)

Input:

Int[(a + b*ArcTanh[c*x^(3/2)])/x^2,x]
 

Output:

-((a + b*ArcTanh[c*x^(3/2)])/x) + 3*b*c*(ArcTanh[c^(1/3)*Sqrt[x]]/(3*c^(1/ 
3)) + (-((Sqrt[3]*ArcTan[(1 - 2*c^(1/3)*Sqrt[x])/Sqrt[3]])/c^(1/3)) - Log[ 
1 - c^(1/3)*Sqrt[x] + c^(2/3)*x]/(2*c^(1/3)))/6 + ((Sqrt[3]*ArcTan[(1 + 2* 
c^(1/3)*Sqrt[x])/Sqrt[3]])/c^(1/3) + Log[1 + c^(1/3)*Sqrt[x] + c^(2/3)*x]/ 
(2*c^(1/3)))/6)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 754

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[-a 
/b, n]], s = Denominator[Rt[-a/b, n]], k, u}, Simp[u = Int[(r - s*Cos[(2*k* 
Pi)/n]*x)/(r^2 - 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x] + Int[(r + s*Cos[(2 
*k*Pi)/n]*x)/(r^2 + 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x]; 2*(r^2/(a*n)) 
 Int[1/(r^2 - s^2*x^2), x] + 2*(r/(a*n))   Sum[u, {k, 1, (n - 2)/4}], x]] / 
; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && NegQ[a/b]
 

rule 851

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = 
 Denominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^ 
n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && 
FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1142

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[(2*c*d - b*e)/(2*c)   Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) 
Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
 

rule 6452

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] : 
> Simp[x^(m + 1)*((a + b*ArcTanh[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m 
+ 1))   Int[x^(m + n)*((a + b*ArcTanh[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))), x 
], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1 
] && IntegerQ[m])) && NeQ[m, -1]
 

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.18

method	result	size
derivativedivides	\(-\frac {a}{x}-\frac {b \,\operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )}{x}-\frac {b \ln \left (\sqrt {x}-\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{2 \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{4 \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{2 \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \ln \left (\sqrt {x}+\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{2 \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{4 \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{2 \left (\frac {1}{c}\right )^{\frac {2}{3}}}\)	\(167\)
default	\(-\frac {a}{x}-\frac {b \,\operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )}{x}-\frac {b \ln \left (\sqrt {x}-\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{2 \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{4 \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{2 \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \ln \left (\sqrt {x}+\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{2 \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{4 \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{2 \left (\frac {1}{c}\right )^{\frac {2}{3}}}\)	\(167\)
parts	\(-\frac {a}{x}-\frac {b \,\operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )}{x}-\frac {b \ln \left (\sqrt {x}-\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{2 \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{4 \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{2 \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \ln \left (\sqrt {x}+\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{2 \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{4 \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{2 \left (\frac {1}{c}\right )^{\frac {2}{3}}}\)	\(167\)

Input:

int((a+b*arctanh(c*x^(3/2)))/x^2,x,method=_RETURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

-a/x-b/x*arctanh(c*x^(3/2))-1/2*b/(1/c)^(2/3)*ln(x^(1/2)-(1/c)^(1/3))+1/4* 
b/(1/c)^(2/3)*ln(x+(1/c)^(1/3)*x^(1/2)+(1/c)^(2/3))+1/2*b/(1/c)^(2/3)*3^(1 
/2)*arctan(1/3*3^(1/2)*(2/(1/c)^(1/3)*x^(1/2)+1))+1/2*b/(1/c)^(2/3)*ln(x^( 
1/2)+(1/c)^(1/3))-1/4*b/(1/c)^(2/3)*ln(x-(1/c)^(1/3)*x^(1/2)+(1/c)^(2/3))+ 
1/2*b/(1/c)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/c)^(1/3)*x^(1/2)-1))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 234 vs. \(2 (102) = 204\).

Time = 0.10 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.65 \[ \int \frac {a+b \text {arctanh}\left (c x^{3/2}\right )}{x^2} \, dx=-\frac {2 \, \sqrt {3} \left (-c^{2}\right )^{\frac {1}{3}} b x \arctan \left (\frac {2 \, \sqrt {3} \left (-c^{2}\right )^{\frac {2}{3}} \sqrt {x} + \sqrt {3} c}{3 \, c}\right ) - 2 \, \sqrt {3} b {\left (c^{2}\right )}^{\frac {1}{3}} x \arctan \left (\frac {2 \, \sqrt {3} {\left (c^{2}\right )}^{\frac {2}{3}} \sqrt {x} - \sqrt {3} c}{3 \, c}\right ) + \left (-c^{2}\right )^{\frac {1}{3}} b x \log \left (c^{2} x - \left (-c^{2}\right )^{\frac {1}{3}} c \sqrt {x} + \left (-c^{2}\right )^{\frac {2}{3}}\right ) + b {\left (c^{2}\right )}^{\frac {1}{3}} x \log \left (c^{2} x - {\left (c^{2}\right )}^{\frac {1}{3}} c \sqrt {x} + {\left (c^{2}\right )}^{\frac {2}{3}}\right ) - 2 \, \left (-c^{2}\right )^{\frac {1}{3}} b x \log \left (c \sqrt {x} + \left (-c^{2}\right )^{\frac {1}{3}}\right ) - 2 \, b {\left (c^{2}\right )}^{\frac {1}{3}} x \log \left (c \sqrt {x} + {\left (c^{2}\right )}^{\frac {1}{3}}\right ) + 2 \, b \log \left (-\frac {c^{2} x^{3} + 2 \, c x^{\frac {3}{2}} + 1}{c^{2} x^{3} - 1}\right ) + 4 \, a}{4 \, x} \] Input:

integrate((a+b*arctanh(c*x^(3/2)))/x^2,x, algorithm="fricas")
 

Output:

-1/4*(2*sqrt(3)*(-c^2)^(1/3)*b*x*arctan(1/3*(2*sqrt(3)*(-c^2)^(2/3)*sqrt(x 
) + sqrt(3)*c)/c) - 2*sqrt(3)*b*(c^2)^(1/3)*x*arctan(1/3*(2*sqrt(3)*(c^2)^ 
(2/3)*sqrt(x) - sqrt(3)*c)/c) + (-c^2)^(1/3)*b*x*log(c^2*x - (-c^2)^(1/3)* 
c*sqrt(x) + (-c^2)^(2/3)) + b*(c^2)^(1/3)*x*log(c^2*x - (c^2)^(1/3)*c*sqrt 
(x) + (c^2)^(2/3)) - 2*(-c^2)^(1/3)*b*x*log(c*sqrt(x) + (-c^2)^(1/3)) - 2* 
b*(c^2)^(1/3)*x*log(c*sqrt(x) + (c^2)^(1/3)) + 2*b*log(-(c^2*x^3 + 2*c*x^( 
3/2) + 1)/(c^2*x^3 - 1)) + 4*a)/x
 

Sympy [F(-1)]

Timed out. \[ \int \frac {a+b \text {arctanh}\left (c x^{3/2}\right )}{x^2} \, dx=\text {Timed out} \] Input:

integrate((a+b*atanh(c*x**(3/2)))/x**2,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.15 \[ \int \frac {a+b \text {arctanh}\left (c x^{3/2}\right )}{x^2} \, dx=\frac {1}{4} \, {\left ({\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {2}{3}} \sqrt {x} + c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right )}{c^{\frac {1}{3}}} + \frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {2}{3}} \sqrt {x} - c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right )}{c^{\frac {1}{3}}} + \frac {\log \left (c^{\frac {2}{3}} x + c^{\frac {1}{3}} \sqrt {x} + 1\right )}{c^{\frac {1}{3}}} - \frac {\log \left (c^{\frac {2}{3}} x - c^{\frac {1}{3}} \sqrt {x} + 1\right )}{c^{\frac {1}{3}}} + \frac {2 \, \log \left (\frac {c^{\frac {1}{3}} \sqrt {x} + 1}{c^{\frac {1}{3}}}\right )}{c^{\frac {1}{3}}} - \frac {2 \, \log \left (\frac {c^{\frac {1}{3}} \sqrt {x} - 1}{c^{\frac {1}{3}}}\right )}{c^{\frac {1}{3}}}\right )} c - \frac {4 \, \operatorname {artanh}\left (c x^{\frac {3}{2}}\right )}{x}\right )} b - \frac {a}{x} \] Input:

integrate((a+b*arctanh(c*x^(3/2)))/x^2,x, algorithm="maxima")
 

Output:

1/4*((2*sqrt(3)*arctan(1/3*sqrt(3)*(2*c^(2/3)*sqrt(x) + c^(1/3))/c^(1/3))/ 
c^(1/3) + 2*sqrt(3)*arctan(1/3*sqrt(3)*(2*c^(2/3)*sqrt(x) - c^(1/3))/c^(1/ 
3))/c^(1/3) + log(c^(2/3)*x + c^(1/3)*sqrt(x) + 1)/c^(1/3) - log(c^(2/3)*x 
 - c^(1/3)*sqrt(x) + 1)/c^(1/3) + 2*log((c^(1/3)*sqrt(x) + 1)/c^(1/3))/c^( 
1/3) - 2*log((c^(1/3)*sqrt(x) - 1)/c^(1/3))/c^(1/3))*c - 4*arctanh(c*x^(3/ 
2))/x)*b - a/x
 

Giac [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.21 \[ \int \frac {a+b \text {arctanh}\left (c x^{3/2}\right )}{x^2} \, dx=\frac {1}{4} \, {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \sqrt {x} + \frac {1}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{{\left | c \right |}^{\frac {1}{3}}} + \frac {2 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \sqrt {x} - \frac {1}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{{\left | c \right |}^{\frac {1}{3}}} + \frac {\log \left (x + \frac {\sqrt {x}}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{{\left | c \right |}^{\frac {1}{3}}} - \frac {\log \left (x - \frac {\sqrt {x}}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{{\left | c \right |}^{\frac {1}{3}}} + \frac {2 \, \log \left (\sqrt {x} + \frac {1}{{\left | c \right |}^{\frac {1}{3}}}\right )}{{\left | c \right |}^{\frac {1}{3}}} - \frac {2 \, \log \left ({\left | \sqrt {x} - \frac {1}{{\left | c \right |}^{\frac {1}{3}}} \right |}\right )}{{\left | c \right |}^{\frac {1}{3}}}\right )} b c - \frac {b \log \left (-\frac {c x^{\frac {3}{2}} + 1}{c x^{\frac {3}{2}} - 1}\right )}{2 \, x} - \frac {a}{x} \] Input:

integrate((a+b*arctanh(c*x^(3/2)))/x^2,x, algorithm="giac")
 

Output:

1/4*(2*sqrt(3)*arctan(1/3*sqrt(3)*(2*sqrt(x) + 1/abs(c)^(1/3))*abs(c)^(1/3 
))/abs(c)^(1/3) + 2*sqrt(3)*arctan(1/3*sqrt(3)*(2*sqrt(x) - 1/abs(c)^(1/3) 
)*abs(c)^(1/3))/abs(c)^(1/3) + log(x + sqrt(x)/abs(c)^(1/3) + 1/abs(c)^(2/ 
3))/abs(c)^(1/3) - log(x - sqrt(x)/abs(c)^(1/3) + 1/abs(c)^(2/3))/abs(c)^( 
1/3) + 2*log(sqrt(x) + 1/abs(c)^(1/3))/abs(c)^(1/3) - 2*log(abs(sqrt(x) - 
1/abs(c)^(1/3)))/abs(c)^(1/3))*b*c - 1/2*b*log(-(c*x^(3/2) + 1)/(c*x^(3/2) 
 - 1))/x - a/x
 

Mupad [B] (verification not implemented)

Time = 10.40 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.55 \[ \int \frac {a+b \text {arctanh}\left (c x^{3/2}\right )}{x^2} \, dx=\frac {b\,c^{2/3}\,\ln \left (\frac {c^{1/3}\,\sqrt {x}+1}{c^{1/3}\,\sqrt {x}-1}\right )}{2}-\frac {a}{x}+\frac {\ln \left (1-c\,x^{3/2}\right )\,\left (b\,x-b\,c^2\,x^4\right )}{2\,x^2-2\,c^2\,x^5}-\frac {b\,\ln \left (c\,x^{3/2}+1\right )}{2\,x}+\frac {b\,c^{2/3}\,\ln \left (\frac {\sqrt {3}+c^{2/3}\,x\,1{}\mathrm {i}-c^{1/3}\,\sqrt {x}\,4{}\mathrm {i}-\sqrt {3}\,c^{2/3}\,x+1{}\mathrm {i}}{2\,c^{2/3}\,x+1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\sqrt {\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\,1{}\mathrm {i}}{2}+\frac {b\,c^{2/3}\,\ln \left (\frac {\sqrt {3}\,c^{2/3}\,x+c^{2/3}\,x\,1{}\mathrm {i}+c^{1/3}\,\sqrt {x}\,4{}\mathrm {i}-\sqrt {3}+1{}\mathrm {i}}{2\,c^{2/3}\,x+1-\sqrt {3}\,1{}\mathrm {i}}\right )\,\sqrt {-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}{2} \] Input:

int((a + b*atanh(c*x^(3/2)))/x^2,x)
 

Output:

(b*c^(2/3)*log((c^(1/3)*x^(1/2) + 1)/(c^(1/3)*x^(1/2) - 1)))/2 - a/x + (lo 
g(1 - c*x^(3/2))*(b*x - b*c^2*x^4))/(2*x^2 - 2*c^2*x^5) - (b*log(c*x^(3/2) 
 + 1))/(2*x) + (b*c^(2/3)*log((3^(1/2) + c^(2/3)*x*1i - c^(1/3)*x^(1/2)*4i 
 - 3^(1/2)*c^(2/3)*x + 1i)/(3^(1/2)*1i + 2*c^(2/3)*x + 1))*((3^(1/2)*1i)/2 
 + 1/2)^(1/2)*1i)/2 + (b*c^(2/3)*log((c^(2/3)*x*1i - 3^(1/2) + c^(1/3)*x^( 
1/2)*4i + 3^(1/2)*c^(2/3)*x + 1i)/(2*c^(2/3)*x - 3^(1/2)*1i + 1))*((3^(1/2 
)*1i)/2 - 1/2)^(1/2))/2
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.82 \[ \int \frac {a+b \text {arctanh}\left (c x^{3/2}\right )}{x^2} \, dx=\frac {2 \sqrt {3}\, \mathit {atan} \left (\frac {2 \sqrt {x}\, c^{\frac {1}{3}}-1}{\sqrt {3}}\right ) b c x +2 \sqrt {3}\, \mathit {atan} \left (\frac {2 \sqrt {x}\, c^{\frac {1}{3}}+1}{\sqrt {3}}\right ) b c x -4 c^{\frac {1}{3}} \mathit {atanh} \left (\sqrt {x}\, c x \right ) b -2 \mathit {atanh} \left (\sqrt {x}\, c x \right ) b c x -4 c^{\frac {1}{3}} a +3 \,\mathrm {log}\left (\sqrt {x}\, c^{\frac {2}{3}}+c^{\frac {1}{3}}\right ) b c x -3 \,\mathrm {log}\left (\sqrt {x}\, c^{\frac {2}{3}}-c^{\frac {1}{3}}\right ) b c x}{4 c^{\frac {1}{3}} x} \] Input:

int((a+b*atanh(c*x^(3/2)))/x^2,x)
 

Output:

(2*sqrt(3)*atan((2*sqrt(x)*c**(1/3) - 1)/sqrt(3))*b*c*x + 2*sqrt(3)*atan(( 
2*sqrt(x)*c**(1/3) + 1)/sqrt(3))*b*c*x - 4*c**(1/3)*atanh(sqrt(x)*c*x)*b - 
 2*atanh(sqrt(x)*c*x)*b*c*x - 4*c**(1/3)*a + 3*log(sqrt(x)*c**(2/3) + c**( 
1/3))*b*c*x - 3*log(sqrt(x)*c**(2/3) - c**(1/3))*b*c*x)/(4*c**(1/3)*x)